Non-conserved currents and gauge-restoring schemes in single W production

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Non-conserved currents and gauge-restoring schemes in single W production

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Ž .

Physics Letters B 479 2000 209–217

Non-conserved currents and gauge-restoring schemes

in single W production

Elena Accomando

a

, Alessandro Ballestrero

b,c,1

, Ezio Maina

b,c a

Texas A & M UniÕersity, College Station, TX , USA

b

INFN, Sezione di Torino, Italy

c

Dipartimento di Fisica Teorica, UniÕersita di Torino, Italy` Received 14 December 1999; accepted 9 March 2000

Editor: R. Gatto

Abstract

We generalize the inclusion of the imaginary parts of the fermionic one-loop corrections for processes with unstable vector bosons to the case of massive external fermions and non conservation of weak currents. We study the effect of initial and final state fermion masses in single W production in connection with the gauge-invariant treatment of the finite-width effects of W and Z bosons, giving numerical comparisons of different gauge-invariance-preserving schemes in the energy

yq y

range of LEP2 and LC for e e ™ e n ud. We do not find significant differences between the results obtained in the_e imaginary part fermion loop scheme and in other exactly gauge preserving methods. q 2000 Elsevier Science B.V. All rights reserved.

PACS: 12.15.-y; 12.15.Ji; 12.15.Lk

Keywords: Electroweak interactions; Gauge invariance; Fermion loop; Single W

1. Introduction

In the past few years a number of papers have discussed the inclusion of weak boson finite-width effects in the theoretical predictions for eq

ey

pro-cesses. A careful treatment is required since these effects are intimately related to the gauge invariance of the theory and any violation of Ward identities can lead to large errors. Even recently a new pro-posal for handling unstable particle processes has

w x

appeared 1 .

1

E-mail: ballestrero@to.infn.it

The most appealing approach used in actual nu-merical computations is in our opinion the

Fermion-Ž . w x

Loop FL scheme 2–5 , which consists in the re-summation of the fermionic one-loop corrections to the vector-boson propagators and the inclusion of all remaining fermionic one-loop corrections, in

particu-w x

lar those to the Yang–Mills vertices. In Ref. 2,3 only the imaginary parts of the loops were included since these represent the minimal set of one–loop contributions which is required for preserving gauge invariance. This scheme will be referred to as the

Ž .

Imaginary Part Fermion-Loop IFL scheme in the

w x

following. In 5 all contributions from fermionic

0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.

Ž .

( ) E. Accomando et al.r Physics Letters B 479 2000 209–217 210

one-loop corrections have been computed. Some ef-fects of light fermion masses in the fermionic loops

w x

have been investigated in 6 .

In this paper we study the effects of external particle fermion masses which imply the non–con-servation of the weak currents which couple to the fermionic loops. These effects have been as yet

q y

w x

neglected for e e ™ 4 f processes: in Ref. 2 and

w x

in the numerical part of Ref. 5 all fermions have

w x

been assumed to be massless, while in Ref. 6 massive matrix elements together with the FL

correc-w x

tions of Ref. 5 were used under the assumption that the currents were conserved. Since our main focus is on gauge invariance, we restrict our attention to the imaginary parts of the fermionic loops, generalizing

w x

the approach of Ref. 2 . The extension of the full FL scheme to the case of massive external fermions is at

w x

present being studied 7 and it will allow to deter-mine the scale of aQE D for single W processes.

We compare the different gauge-restoring schemes

y q y

Ž .

in e e ™ e n ud CC20 which, in addition to the_e

y q y

Ž .

usual diagrams of e e ™ m n ud CC10 , requiresm

all diagrams obtained exchanging the incoming eq

with the outgoing ey

. These contributions become dominant for u ™ 0 because of the t-channel g_e propagator. The CC20 four fermion events with e lost in the pipe are often referred to as single W production, and are relevant for triple gauge studies and as background to searches. For recent reviews

w x

see Refs. 9 . Since the t-channel g propagator diverges at u s 0 in the m ™ 0 limit, fermione e

masses have to be exactly accounted for. Moreover, the apparent ty2

behaviour is reduced to ty1

by gauge cancellations. This implies that even a tiny violation of gauge conservation can have dramatic

w x

effects, as e.g. discussed in Ref. 2,8 , and the use of some gauge conserving scheme is unavoidable.

Two different strategies have been used:

Im-w x

proved Weiszacker-Williams 10 implemented in

w x

WTO 11 and completely massive codes. In the first case one separates the 4 t-channel photon diagrams, evaluates them analytically in the equivalent photon approximation taking into account the complete de-pendence on all masses, and then adds the rest of diagrams and the interference between the two sets in the massless approximation. In the fully massive

w x w x

MC numerical approachCOMPHEP 12 ,GRC4F 13 ,

w x w x

KORALW 14 , WPHACT 15 and recently also the

y

Ž .qŽ .

Fig. 1. The four diagrams of the process e p e1 k ™1

y

Ž . Ž . Ž . Ž .

e p n k2 e 2 u p d pu d which are considered in this paper.

w x w x

two new codes NEXTCALIBUR 16 and SWAP 17 have compared their results and found good

agree-w x2

ment 9 among themselves and withWTO.

Ž .

In the following we first discuss the issue of U 1 gauge invariance in single W production with non conserved weak currents. We then give the expres-sion of all required contributions to the vertex cor-rections in the IFL scheme. Finally we present com-parisons between the IFL and other gauge-preserving schemes which have been employed in the literature and study the relevance of neglecting current non conservation in the energy range of LEP2 and LC.

2. Gauge invariance

We choose to work in the unitary gauge. In this case, the relevant set of Feynman diagrams which become dominant for u ™ 0 coincides with thosee

w x

discussed in Ref. 2 . They are shown in Fig. 1. For ease of comparison we follow closely the notation of

w x_{2 . The corresponding matrix element MM is given}

by Qe m m m MMs MM J ,m J s 2u p

Ž

2

.

g u p

Ž

1

.

, q 4 m m MM s

_Ý

MMi

Ž .

1 is1 2

More details can be found in the homepage of the LEP2 MC Workshop http:// www.ph.unito.it / ~giampier / lep2.html

where M M₁ms_{QW PW}

_Ž

_pq2

_.

_PW

_Ž

_py2

_.

_Va bm

_Ž

_{p ,y p ,y qq y}

_.

= Dr p Ds p MM0 ,

Ž

.

Ž

.

a q b y sr k

u

q_q

u

y_m 1 e m 2 2 m M M s 4 iQ g P_{2 e w W}

_Ž

pq

_.

Õ k

_Ž

₁

_.

g ₂ 2 k q q y_m

Ž

1

.

e a r =_{g P Õ k}

_Ž

_.

_{u p g P Õ p}

_Ž

_.

_Ž

_.

_D

_Ž

_p

_.

_, L 2 u r L d a q m 2 2 M M s y4 iQ g P3 u w W

_Ž

py

_.

u p

Ž

u

.

p

u

y_q

u

q_m u u m b =g 2 2g P Õ pL

Ž

d

.

p y q y_m

Ž

u

.

u s = Õ k g P Õ k

Ž

1

.

s L

Ž

2

.

Db

Ž

py

.

, m 2 2 M M s y4 iQ g P4 d w W

Ž

py

.

u p

Ž

u

.

q

u

y_p

u

q_m d d b m =_{g P g Õ p}

_Ž

_.

L 2 ₂ d p y q y_m

Ž

d

.

d s = Õ k g P Õ k

Ž

1

.

s L

Ž

2

.

Db

Ž

py

.

, 0 2 M M ' 4 ig Õ k g P Õ ksr w

Ž

1

.

s L

Ž

2

.

u p g P Õ p

Ž

u

.

r L

Ž

d

.

, 2

Ž .

1_Ž 5_. where P 'L 2 1 y g and p s p q p , p s k y k , q s p y p ,q u d y 1 2 1 2

Ž .

3 y1 ₂ P

_{Ž .}

s s_{s y M q ig}

_{Ž .}

_{s ,}

_{Ž .}

₄ W W W D_ab

_Ž

p s g

_.

_aby_{p pa} br_{K p}

_Ž

2

_.

_.

_{Ž .}

₅

MW is the W mass and gW denotes the imaginary part of the inverse W propagator. At tree level,

Ž 2. 2

K p s_{MW but the resummation of the imaginary}

parts of higher order graphs modifies the lowest order expression of K in addition to generate a finite width. The charged weak coupling constant gw is

2 2

_'

given by g s M G r 2 , while Q is the electricw W F i

charge of particle i, and

m3 m m m_{1 2 3} m m_{1 2} V

_Ž

p , p , p_{1 2 3}

_.

s

_Ž

_{p y p1 2}

_.

_g m_{1 m m} 2 3 q

_Ž

_{p y p2 3}

_.

_g m_{2 m m} 3 1 q

_Ž

_{p y p}

_.

_{g .}

_{Ž .}

₆ 3 1

The conservation of electromagnetic current requires

qmMM s 0 .m

Ž .

7

Any small violation of this relation will be amplified by a huge factor and will lead to totally wrong

w x

predictions for almost collinear electrons 2,8 .

Mul-m

Ž .

tiplying q into the four diagrams of Eq. 2 , we obtain W ' qm M M s MM p2 _yp2 _{Q P p}2 _{P p}2



Ž

.

Ž

.

Ž

.

m 0 q y _{W W} q _W y q_{Q Pe W}

_Ž

_pq2

_.

y

_Ž

_{Q y Qd u}

_.

_PW

_Ž

_py2

_.

₄

y MM Q P p2 _{P p}2



Ž

.

Ž

.

qq W W q W y = 1 y p2_{rK p}2 _{qQ P p}2 _{rK p}2

4

Ž

.

Ž

.

Ž

.

Ž

y q

.

_{e W} q q q MM Q P p2 _{P p}2



Ž

.

Ž

.

yy _{W W} q _W y = 1 y p2_{rK p}2

Ž

.

Ž

q y

.

q

_Ž

_{Q y Q}

_.

_P

_Ž

_p2

_.

r_{K p}

_Ž

2

_.

₄

d u W y y q MMyq

_½

QW PW

_Ž

pq2

_.

PW

_Ž

py2

_.

y1 y1 2 2 =p P p_{y q}

_ž

K p

_Ž

_y

_.

y_{K p}

_Ž

_q

_.

_/

5

_,

_{Ž .}

₈ where MM ' MM0 ga b , MM ' MM0 pa pb ,

Ž .

9 0 a b qq a b q q MMyy' MMa b0 pyapyb , MMyq' MMa b0 pyapqb.

Ž

10

.

Ž .

Using Q s Q s Q y Q s y1 and Eq. 4 weW e d u

have W s i MM P p2 P p2 g _p2 _yg p2

Ž

.

Ž

.

Ž

Ž

.

Ž

.

.

0 W q W y W q W y q MM P p2 _{P p}2



Ž

.

Ž

.

qq W q W y = 1 y M2_{yig p}2 _{rK p}2

4

Ž

.

Ž

.

Ž

.

Ž

W W y q

.

y MM P p2 _{P p}2



Ž

.

Ž

.

yy W y W q = 1 y M2_{yig p}2 _{rK p}2

4

Ž

.

Ž

.

Ž

.

Ž

W W q y

.

y MMyq

½

P_W

Ž

pq2

.

P_W

Ž

py2

.

y1 y1 2 2 =p P p_{y q}

_ž

K p

_Ž

_y

_.

y_{K p}

_Ž

_q

_.

_/

₅

_.

_Ž

₁₁

_.

Current conservation is therefore violated unless

2 2

gW

Ž

pq

.

sg_W

Ž

py

.

'g_W

Ž

12

.

2 2 2

K p

_Ž

_q

_.

s_{K p}

_Ž

_y

_.

s_{M y ig}

_Ž

₁₃

_.

W W

It should be mentioned that all effects due to the non conservation of the currents which couple to the W and Z bosons are contained in the last three terms of

Ž . Ž .

Eq. 8 and Eq. 11 which would be zero if the currents were conserved.

( ) E. Accomando et al.r Physics Letters B 479 2000 209–217 212

The most naive treatment of a Breit-Wigner reso-nance uses a fixed width approximation, with

g s M GW W W.

Ž

14

.

Ž . Ž .

Eqs. 11 – 13 show that in this case there is no violation of electromagnetic current conservation. In the unitary gauge this corresponds to adding the same imaginary part, yiM G , to M2 _{both in the}

W W W

denominator and in the pmpn term of the W

propaga-Ž w x _.

tor see e.g. 3 and references therein . We have verified numerically that neglecting to modify the latter leads to large errors already at 800 GeV. A similar approach, in which all weak boson masses squared M2, B s W, Z are changed to M2y_ig

ev-B B B

erywhere, including in the definition of the weak

w x

mixing angle, has in fact been suggested 18 as a

Ž . Ž .

mean of preserving both U 1 and SU 2 Ward iden-tities in the Standard Model.

The fixed-width approximation cannot however be justified from field theory. Indeed, propagators with space-like momenta are real and cannot acquire a finite width in contradiction to the fixed-width scheme.

w x

As discussed in Ref. 2 , the simplest way to restore gauge-invariance in a theoretically satisfying fashion is the addition of the imaginary parts of one-loop fermionic vertex corrections, shown in Fig. 2, which cancel the imaginary part in the Ward identities. The cancellation is exact as long as all fermion loops, both in the vertices and in the propa-gators, are computed in the same approximation. In particular we can consistently neglect fermion masses in the loops, if we use for the W width the tree–level expression for the decay of an on-shell W to mass-less fermions G M3 F W G sW

_Ý

Nf ,

Ž

15

.

'

6p 2 doublets Ž

involving a sum over all fermion doublets with Nf 1

.

or 3 colours.

Fig. 2. The extra fermionic diagrams needed to cancel the gauge-breaking terms.

Fig. 3. First order contribution to the inverse W propagator. The fermions in the loop are assumed to be massless.

The vertex corrections are given by

i m 0 2 2 2 MM s₅ MMrs PW

_Ž

pq

_.

P_W

_Ž

py

_.

g_w 16p = _{N Q y Q D}r p

Ž

.

Ž

.

Ý

f d u a q doublets = Ds p Za bm , 16

Ž

.

Ž

.

b y where 1 r

u

y_q

u

1 a bm m b a Z s

_H

_{dV Tr r}

u

₁g g r

u

₂g 2 2p

Ž

r y q1

.

17

Ž

.

is the imaginary part of the triangle insertions. The momenta r1 and r2 are the momenta of the cut fermion lines with p s r q r . The expressionq 1 2

Za bm satisfies the three Ward identities:

8 a bm a b 2 a b a bm q Z q s ym 3

Ž

p p y p gq q q

.

, Z p s 0 ,a 8 a bm y m a 2 m a Z p s qb 3

Ž

p p y p gq q q

.

.

Ž

18

.

Because of the anomaly cancellation we have no explicit contributions from the part containing g5. Attaching the photon momentum qm to the sum of the diagrams MM5m gives

p2 q m 2 2 W_add' q M_mM s y i M₅ M P_{0 W}

_Ž

pq

_.

P_W

_Ž

py

_.

G_W M_W G_W 2 2 q_{i MMqqPW}

_Ž

_pq

_.

_PW

_Ž

_py

_.

MW G _p2 W q 2 2 q_{i MMyyPW}

_Ž

_pq

_.

_PW

_Ž

_py

_.

2 MW K p

_Ž

y

_.

G_W p P pq y 2 2 y_{i MMyqP}

_Ž

_p

_.

_P

_Ž

_p

_.

W q _W y ₂ MW K p

_Ž

y

_.

19

Ž

.

Ž .

where we used the Ward identity of Eq. 18 and the

Ž .

Assum-Ž 2.

ing g_W p_y s_{0 as required by field theory, the} Ž .

extra diagrams restore U 1 gauge invariance pro-vided pq2 2 g_W

_Ž

_pq

_.

sG_{W ,}

_Ž

₂₀

_.

MW y1 G_W 2 2 K p

_Ž

_q

_.

s_{MW 1 q i ,}

_Ž

₂₁

_.

ž

MW

/

K p2 _sM2 _{. 22}

Ž

.

Ž

y

.

_W

This result may be surprising but it is actually the correct field theoretical resummation, in the unitary gauge, of the imaginary part of the fermionic

one-w x

loop contributions 3 , shown in Fig. 3, which is transverse if we consider only massless fermions Im P

_Ž

_Wmn

_.

s

_Ž

_gmny_pm_pnr_p2

_.

P_W,

_Ž

₂₃

_.

with G_W 2 P s pW .

Ž

24

.

MW

If, suppressing indices for simplicity, we define

| ' gmn, D ' gmny_pm_pnr_M2 _{and T ' g}mny

pmpnr_p2 _{we have DT s TD s T and T}2s_{T. The}

usual Dyson series for the resummation of the imagi-nary part P of one–loop corrections reads:

y_iD y_iD y_iD S s q

_Ž

_TP

_.

qPPP 2 2 2 2 2 2 p y M p y M p y M y_{iD i P} s | q

_Ž

_{| y T}

_.

_. 2 2

ž

2 2

/

p y M q i P p y M 25

Ž

.

More explicitly y_{i p}m_pn _{i P} mn mn S s _g y _{1 q .} 2 2

½

2

ž

2

/

5

p y M q i P M p 26

Ž

.

Hence the introduction of a finite width for s-channel virtual W ’s which is required even in tree level calculations has to be associated with a corre-sponding modification of the pmpn term.

3. Form factors for the vertex corrections

We report in this section the analytic expression of Za bm which is needed for actual computations in the FL scheme. Parametrizing Za bm as follows

Za bms_pa_pb_pm_{f q q}a_pb_pm_{f q p}a_qb_pm_f q q q 1 q q 2 q q 3 q_pqa_pqb_qm_{f q q4} a_qb_pqm_{f q q5} a_pqb_qm_f6 q_pqa_qb_qm_{f q q7} a_qb_qm_{f q g8} a b_pqm_f9 q_ga b_qm_{f q g}bm_pa_{f q g}bm_qa_f 10 q_{11 12} q_gam_pqb_{f q g}am_qb_f

_Ž

₂₇

_.

13 14 we find 2 2 2 p q pq y 2 2 2 2 2 f s 1601 ₃

½

f y6 p q p y 2 p q p0 q y

_Ž

q y

_.

l 2 4 4 = p P p

_Ž

_{y q}

_.

q_{2 p q p}

_Ž

_{q y}

_.

_{p P py q} 2 2 pq

Ž

p P py q

.

_{2 2 4} q₁₀ q_{20 p p q 10 pq y q} 2 q 2 ₂ 6 2 6 4 y

_Ž

_{3 p q p P pq}

_Ž

_{y q}

_.

_{p q py y}

_.

q_{2 py} 2 q 2 2 4 2 2 l _{q py py p qq} 2 q _f0 y q y y_{6 pq} 2 2 2

ž

10 pq pq py pq4 116 2 2 q y_{10 q y 6 py} 2

/

3 py p4 _p2 _{139 p}2 y y q y₂ q₂ q 2 2 2 ₃ 2 q pq pq q 2 4 2 pq 20 pq 67 py q₁₄ y q 2 ₃ 2 2 ₃ 2 py q py q 2 l 1 1 7 q y_f q q 0

ž

2 2

/

2 2 10 pq py 3 p qq 2 19 q q _,

_Ž

₂₈

_.

2 2 2 2

5

3 p pq y 3q py

( ) E. Accomando et al.r Physics Letters B 479 2000 209–217 214 2 2 2 p q pq y 2 2 2 2 4 f s y160_{2 3}

½

f₀ p q p q p qq y q l 2 4 4 2 6 2 y_{q p q 2 q p y qy y} q_{2 p q P pq y} f l0 2 4 2 q_{6 q q P p q 4 q y 2 q P p p qy y y} 20 p2_q2 _q4 q 2 2 2 = 2 q_{3 p q 17q y 2q} y_{3 py} 2 2

ž

py py

/

l 2 2 q

_Ž

_{11q q 13q P p y py y}

_.

2 15 py l2 2 q

_Ž

_{1 q 3 f q0}

_.

_,

_Ž

₂₉

_.

2 2

5

60 q py 2 2 2 p q pq y 2 2 2 f s 160_{3 3}

½

f 3 p q p₀ q y l 2 2 2 4 q

_Ž

_{3 p y pq y}

_.

_Ž

_{p P py q}

_.

y_{2 p p P pq y q} 2 2 6 pq

Ž

p P py q

.

pq 2 2 4 y₈ y_{8 p p y 8 p q 4q y q} 2 2 q q 2 2 p P p p

Ž

y q

.

y q₄ 2 q 2 2 4 l _{p qq pq} 13 2 2 2 q _f0 q_{3 p yq} q _{q q 3 py} 2 2 2

ž

/

10 py py p2 _{40 p}2 q q y_{20 y 32} y 2 ₃ 2 q py 4 2 pq py q₆ y₄ 2 2 2 q py q 2 l _{1 2} q _f0 q 2 2

ž

/

20 pq py 1 1 34 y y y _,

_Ž

₃₀

_.

2 2 2 2 2 2

5

p qq p pq y 3q py 2 2 2 p q pq y 2 2 2 2 4 2 4 f s 160_{5 3}

½

f 3 p q p q p q y q p₀ q y q y l 4 2 6 q_{2 q p y qy} q_{6 pq}2_{q P p q 6 qy} 2_{q P p q 4 qy} 4y_{2 q P p py y}2 f l p2_q2 _q4 0 q 2 2 2 q ₂ q_{7p q 21q y 2q} q_py 2 2

ž

/

20 py py l _{2 q P p 11 q}2 _{q P p} y y q y_{3 q} q q₅ 2 2 2

ž

/

5 3 q 3 py py l2 2 q

_Ž

_{1 q 3 f q}

_.

_,

_Ž

₃₁

_.

0 2 2

5

60 q py 2 2 2 p q pq y l 2 f s 16 f q p P p q q_{2 q P p} 9 ₂

½

0 y q q 2 l l 2 q

_Ž

_{4 p P q y 3qq}

_.

_Ž

₃₂

_.

2 2

5

6 q py 2 2 2 p q pq y 2 f s 1611 ₂

½

f0 q p P py q l p P py q 1 ql y q q_{2 p P qq} 4 2

ž

2 pq

/

1 1 2 q_{l p P qq} q y _, 2 2 2 2 2 2

5

ž

2 p qq 2 p pq y 3q py

/

33

Ž

.

2 2 2 p q p_{q y} l 2 f s 1612 ₂

½

f yp q P p q0 q y 2 l l 2 2 y_{2 p qq}

_Ž

_{6 q P p y py q}

_.

_,

_Ž

₃₄

_.

2 2

5

6 q py 2 2 2 p q pq y _{2 2} f s 1613 ₂

½

f y7q p P q q 4 p P q0 q

Ž

q

.

l 2 p P qq q 3 2 2 4 y_{3q p q 3q q l y qy} y 4 2 2

ž

2 py 2 py

/

2 14

Ž

p P py q

.

8 2 2 22 q

_Ž

_{p y qy}

_.

y _{p P p qy q} 3 3 _{3 py}2 l q _{q P py ,}

_Ž

₃₅

_.

2 2

5

6 q py

2 2 2 p q pq y 2 2 f s 16_{14 2}

½

f 7p P qq y p P qp₀ q q y l 2 _{2 2 4} y_{4 p P q}

_Ž

_q

_.

q_{3q p y 3qy} 2 p P qq q 1 ql y q 4 2 2

ž

2 py 2 py

/

2 14

Ž

p P py q

.

8 2 2 16 y

_Ž

_{p y qy}

_.

q _{p P p yy q} 3 3 _{3 py}2 l 2 q

_Ž

_{4 p y 5 p P py y q}

_.

_,

_Ž

₃₆

_.

2 2

5

6 q py with

'

2 2 p P q q l

Ž

y

.

f s y₀ ln ,

'

l

ž

2 p P q y l

Ž

y

.

'

/

2 2 2 l ' 4 p P q

_Ž

_y

_.

y_{4 p q .y}

_Ž

₃₇

_.

The form factors f , f , f , f , f_{4 6 7 8 10} do not con-tribute in CC20 processes because the electron cur-rent is conserved. If we assume that all curcur-rents which couple to the fermion loops are conserved we have Za bm_s qa qb pm c q gbm qa c q gam qb c q _{0 1 2} q_ga b_pqm_{c ,}

_Ž

₃₈

_.

3 c s f q f ,_{0 2 5} c s f_{1 12}, c s f q f_{2 13 14}, c s f ,3 9

Ž

39

.

w x

where the c agree with the results of Ref. 2 ._i

I H I

4. Applications to the process e e ™ e n ud:_e numerical effects in a physically relevant case study

In the following we present numerical results obtained with the IFL scheme and make comparisons with those obtained with other gauge-preserving ap-proaches. The following schemes are considered in our analysis:

Ž .

Imaginary-part FL scheme IFL : The imaginary part

Ž . Ž .

of the fermion-loop corrections Eq. 27 – 36 are used. The fermion masses are neglected in the loops but not in the rest of the diagrams.

Ž .

Fixed width FW : All W-boson propagators are given by pm pn mn g y 2 M y i G MW W W .

Ž

40

.

2 2 p y M q i G MW W W

This gives an unphysical width for p2- 0, but Ž .

retains U 1 gauge invariance.

Ž .

Complex Mass CM : All weak boson masses squared M2_{, B s W, Z are changed to M}2_{yig ,}

B B B

including when they appear in the definition of the weak mixing angle. This scheme has the

Ž . Ž .

advantage of preserving both U 1 and SU 2

w x

Ward identities 18 .

Ž . y q

Overall scheme OA : The diagrams for e e ™

y

e n ud can be split into two sets which aree

Ž .

separately gauge invariant under U 1 . In the

pre-w x

sent implementation of OA 19 , t-channel dia-grams are computed without any width and are

Ž 2 2. Ž 2 2 .

then multiplied by q y M r _{q y M q iMG}

where q, M and G are the momentum, the mass and the width of the possibly-resonant W-boson.

Ž .

This scheme retains U 1 gauge invariance at the expenses of mistreating non resonant terms. In order to assess the relevance of current

non-y q y

conservation in process e e ™ e n ud we have_e also implemented the imaginary part of the fermion-loop corrections with the assumption that all currents which couple to the fermion-loop are conserved. In

Ž . Ž .

this case Eq. 27 – 36 reduce to those computed in

w x_{2 . Notice that the masses of external fermions are}

nonetheless taken into account in the calculation of

Ž .

the matrix elements. This scheme violates U 1 gauge invariance by terms which are proportional to the

w x

fermion masses squared, as already noted in Ref. 6 . However they are enhanced at high energy by large factors and can be numerically quite relevant. This scheme will be referred to as the imaginary-part FL

( )

scheme with conserved currents IFLCC in the fol-lowing.

In the comparisons among the different codes mentioned in the introduction, COMPHEP and

WPHACTused the OA scheme,KORALWandGRC4F

mn _{w x}

the L transform method of Ref. 8 , NEXTCAL

( ) E. Accomando et al.r Physics Letters B 479 2000 209–217 216

all schemes described above have been implemented in the new version ofWPHACT in which all massive matrix elements have been added to the old massless

Ž .

ones. In particular the IFL contributions in Eq. 27 –

Ž36 have been introduced. In this way, the same.

matrix elements, phase spaces and integration rou-tines are used in all instances.

If not stated otherwise we apply the following cuts:

M ud ) 5 GeV ,

Ž

.

E ) 3 GeV,u E ) 3 GeV,d

cos u

Ž

e

.

) .997

Ž

41

.

We have produced numerical results for ey

eq

™

y 2 _Ž

e n ud in the small space-like q_e g collinear

elec-.

tron region where we expect gauge-invariance is-sues to be essential. We have not included in our

Ž .

computations Initial State Radiation ISR , in order to avoid any additional uncertainty in these compar-isons among different gauge restoring schemes. In Table 1 we give the cross sections for CC20 at Lep 2 and LC energies. In Table 2 we give the cross sections for CC20 at E s 800 GeV with slightly modified selections. With all other cuts at their standard values, in the second column the electron scattering angle is not allowed to be larger than 0.1 degree while in the third column the invariant mass of the ud pair is required to be greater than 40 GeV. The IFL, FW, CM and OA schemes agree within 2 s in almost all cases. The IFLCC scheme agrees with all other ones at Lep 2 energies but already at 800 GeV it overestimates the total cross section by about 6%. At 1.5 TeV the error is almost a factor of two. The results in Table 2 show that the discrepancy between the IFLCC scheme and all the others de-creases slightly to 5.6 % if larger masses of the ud pair are required. If instead smaller electron

scatter-Table 1

q y y

Cross sections for the process e e ™ e n ud for various gaugee

restoring schemes

190 GeV 800 GeV 1500 GeV

Ž . Ž . Ž . IFL 0.11815 13 1.6978 15 3.0414 35 Ž . Ž . Ž . FW 0.11798 11 1.6948 12 3.0453 41 Ž . Ž . Ž . CM 0.11791 12 1.6953 16 3.0529 60 Ž . Ž . Ž . OA 0.11760 10 1.6953 13 3.0401 23 Ž . Ž . Ž . IFLCC 0.11813 12 1.7987 16 5.0706 44 Table 2 q y y

Cross sections for the process e e ™ e n ud at E s800 GeVe

for various gauge restoring schemes and different cuts

Ž . Ž . cos u ).997e u - 0.18e M ud ) 40 GeV Ž . Ž . Ž . IFL 1.6978 15 1.1550 15 1.6502 15 Ž . Ž . Ž . FW 1.6948 12 1.1538 21 1.6480 13 Ž . Ž . Ž . CM 1.6953 16 1.1533 14 1.6520 10 Ž . Ž . Ž . OA 1.6953 13 1.1537 12 1.6523 12 Ž . Ž . Ž . IFLCC 1.7987 16 1.2600 22 1.7424 21

ing angle are allowed the discrepancy increases to about 9%. This is a consequence of the fact that in the collinear region the neglected terms, proportional to the fermion masses, are enhanced by factors of order OO m2g p2 _{r M}2_m2 _{which can become}

Ž

.

Ž

.

Ž

f W W e

.

very large at high energy even for typical light fermion masses.

We conclude then that, even in the presence of non–conserved currents i.e. of massive external fermions, the FW, CM and OA calculations give predictions which are in agreement, within a few per mil, with the IFL scheme. This agreement with the results of a fully self-consistent approach justifies from a practical point of view the ongoing use of the FW, CM and OA schemes. It should be remarked that for massless fermions it has been shown that at high energies, for the total cross section of the

y q y

process e e ™ m n ud the full FL scheme deviatesm

from the FW scheme and the IFL scheme by about

w x

2% at 1 TeV increasing to about 7% at 10 TeV 5 mainly because of the running of the couplings. As a consequence, it appears likely that calculations per-formed in the IFL scheme with running couplings would be able to reproduce the complete FL results with sufficient accuracy for most practical purposes. Hitherto missing higher order QCD and bosonic contributions could still conceivably produce signifi-cant corrections.

5. Conclusions

The Imaginary Part Fermion-Loop scheme,

intro-w x

duced in Ref. 2 for the gauge-invariant treatment of the finite-width effects of W and Z bosons, has been generalized so that it could be applied to processes with massive external fermions. This involves the

Dyson resummation of higher order imaginary con-tributions to the propagator which implies, in the unitary gauge, a modification of the pmpn term in the numerator. From a numerical point of view we find no significant difference between the IFL scheme and the FW, CM or OA schemes in the region most

Ž .

sensible to U 1 gauge invariance.

Acknowledgements

This research has been partly supported by NSF Grant No. PHY-9722090 and by MURST. We wish to thank G. Passarino for several discussions on gauge invariance and related issues. We also grate-fully acknowledge the exchange of information and comparisons with other groups and in particular with E. Boos, M. Dubinin, S. Jadach and R. Pittau.

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